Every group is the automorphism group of a rank-3 matroid

Every group is the automorphism group of a rank-3 extension of a rank-3 Dowling geometry.Partially supported by The George Washington University UFF grant.Partially supported by the National Security Agency under grant MDA904-91-H-0030.     

Every finite group is the automorphism group of some perfect code

In this paper we prove the every finite group is isomorphic to the full automorphism group of some perfect binary single error correcting code.     

Every compact group arises as the outer automorphism group of a II1 factor

We show that any compact group can be realized as the outer automorphism group of a factor of type II₁. This has been proved in the abelian case by Ioana, Peterson and Popa [A. Ioana, J. Peterson, S. Popa, Amalgamated free products of w-rigid factors and calculation of their symmetry group, math.OA/0505589, Acta Math., in press] applying Popa's deformation/rigidity techniques to amalgamated free product von Neumann algebras. Our methods are a generalization of theirs.     

The automorphism tower problem II

We prove that the automorphism tower of every infinite centreless groupG of cardinality κ terminates in less than (2^κ)⁺ steps. We also show that it is consistent withZFC that the automorphism tower of every infinite centreless groupG of regular cardinality κ actually terminates in less than 2^κ steps. Research partially supported by NSF Grants.     

The pascal automorphism has a continuous spectrum

In this paper we describe the Pascal automorphism and present a sketch of the proof that its spectrum is continuous on the orthogonal complement of the constants.     

Every poset has a central element

It is proved that there exists a constant , such that in every finite partially ordered set there is an element such that the fraction of order ideals containing that element is between δ and 1−δ. It is shown that δ can be taken to be at least (3−log₂ 5)/40.17. This settles a question asked independently by Colburn and Rival, and Rosenthal. The result implies that the information-theoretic lower bound for a certain class of search problems on partially ordered sets is tight up to a multiplicative constant.     

Every Set has a Least Jump Enumeration

Given a computably enumerable set W, there is a Turing degreewhich is the least jump of any set in which W is computablyenumerable, namely 0'. Remarkably, this is not a phenomenonof computably enumerable sets. It is shown that for every subsetA of N, there is a Turing degree, c'_µ(A), which is theleast degree of the jumps of all sets X for which A is . In addition this result providesan isomorphism invariant method for assigning Turing degreesto certain torsion-free abelian groups.     

On the outer automorphism group of a hyperbolic group

LetG be a one-ended, word-hyperbolic group. Let Γ be an irreducible lattice in a connected semi-simple Lie group of rank at least 2. Ifh: Γ→Out(G) is a homomorphism, then Im(h) is finite. Dedicated to Professor Takushiro Ochiai for his sixtieth birthday     

On the automorphism group of a tree

It is shown that H = Γ(T)_v is normal in G = Γ(T_v) for any tree T and any vertex v, if and only if, for all vertices u in the neighborhood N of v, the set of images of u under G is either contained in N or has precisely the vertex u in common with N and every vertex in the set of images is fixed by H. Further, if S is the smallest normal subgroup of G containing H then $\text{G}\text{S}$ is the direct product of the wreath products of various symmetric groups around groups of order 1 or 2. The degrees of the symmetric groups involved depend on the numbers of isomorphic components of T_v and the structure of such components.     

The automorphism group of a coded system

We give a general construction of coded systems with an automorphism group isomorphic to where is any preassigned group which has a ``continuous block presentation' (the isomorphism will map the shift to . Several applications are given. In particular, we obtain automorphism groups of coded systems which are abelian, which are finitely generated and one which contains . We show that any group which occurs as a subgroup of the automorphism group of some subshift with periodic points dense already occurs for some synchronized system.

     

The automorphism group of a circle geometry

The following is proved: Theorem. Let n be an integer, n ? 2. Let F be a field with at least Max(n, 3) distinct elements and let K be a separable extension field of dimension n over F. Then the group of all automorphisms of the n-dimensional circle geometry CG(F, K) is the group induced on the 1-dimensional vector subspaces over K of the underlying 2-dimensional vector space V over K by the semi-linear permutations of V over K for which the associated automorphisms of K induce automorphisms of F.With the hope of broadening the interest in the foundations of geometry, the author attempts to give anough detail so that, without consulting the literature, any attentive algebraist can understand clearly both the content of the theorem and the pattern of its proof and can identify easily any gaps in his knowledge which might prevent him from viewing this paper as a contribution to geometry. Adequate references are given.     

On the automorphism group of a cone

Let V be a real finite dimensional vector space, and let C be a full cone in C. In Sec. 3 we show that the group of automorphisms of a compact convex subset of V is compact in the uniform topology, and relate the group of automorphisms of C to the group of automorphisms of a compact convex cross-section of C. This section concludes with an application which generalizes the result that a proper Lorentz transformation has an eigenvector in the light cone. In Sec. 4 we relate the automorphism group of C to that of its irreducible components. In Sec. 5 we show that every compact group of automorphisms of C leaves a compact convex cross-section invariant. This result is applied to show that if C is a full polyhedral cone, then the automorphism group of C is the semidirect product of the (finite) automorphism group of a polytopal cross-section and a vector group whose dimension is equal to the number of irreducible components of C. An example shows that no such result holds for more general cones.